Definitions

Let us work in the context of universal algebra, so an algebra is a setXX equipped with a family of functionsfi:Xni→Xf_i\colon X^{n_i} \to X (where each arity?nin_i is a cardinal number) that satisfy certain equational identities (which are irrelevant here). As usual, a subalgebra of XX is a subsetSS such that fi(p1,…,pni)∈Sf_i(p_1,\ldots,p_{n_i}) \in S whenever each pk∈Sp_k \in S.

A subsetAA of XX is open (relative to ≠\ne) if p∈Ap \in A or p≠qp \ne q whenever q∈Aq \in A. An antisubalgebra of XX is an open subset AA such that some pk∈Ap_k \in A whenever fi(p1,…,pni)∈Af_i(p_1,\ldots,p_{n_i}) \in A. By taking the contrapositive?, we see that the complement of AA is a subalgebra SS; then AA may be recovered as the ≠\ne-complement of SS (the set of those pp such that p≠qp \ne q whenever q∈Sq \in S). However, we cannot start with an arbitrary subalgebra SS and get an antisubalgebra AA in this way, as we cannot (in general) prove openness. (We can take the antisubalgebra generated by the ≠\ne-complement of SS, as described below, but its complement will generally only be a superset of SS.)

Examples

The empty subset of any algebra is an antisubalgebra, the empty antisubalgebra or improper antisubalgebra, whose complement is the improper subalgebra (which is all of XX). An antisubalgebra is proper if it is inhabited; the ability to have a positive definition of when an antisubalgebra is proper is a significant motivation for the concept.

If AA is an antisubalgebra and cc is a constant (given by an operation X0→XX^0 \to X or a composite of same with other operations), then p≠cp \ne c whenever p∈Ap \in A. If there are only Kuratowski-finitely many constants (which is needed to prove openness), we define the trivial antisubalgebra to be the subset of those elements pp such that p≠cp \ne c for each constant cc (the ≠\ne-complement of the trivial subalgebra?). In general, we may also take the trivial antisubalgebra to be the union of all antisubalgebras, although this is not predicative.

Instead of subgroups, use antisubgroups. In detail, AA is an antisubgroup if p≠1p \ne 1 whenever p∈Ap \in A, p∈Ap \in A or q∈Aq \in A whenever pq∈Ap q \in A, and p∈Ap \in A whenever p−1∈Ap^{-1} \in A. An antisubgroup AA is normal if pq∈Ap q \in A whenever qp∈Aq p \in A. The trivial antisubgroup is the ≠\ne-complement of {1}\{1\}.

Instead of ideals (of commutative rings), use antiideals (and we also have left and right antiideals of general rings). In detail, AA is an antiideal if p≠0p \ne 0 whenever p∈Ap \in A, p∈Ap \in A or q∈Aq \in A whenever p+q∈Ap + q \in A, and p∈Ap \in A whenever pq∈Ap q \in A. It follows that an antiideal AA is proper iff 1∈A1 \in A. AA is prime if it is proper and pq∈Ap q \in A whenever p∈Ap \in A and q∈Aq \in A; AA is minimal if it is proper and, for each p∈Ap \in A, for some qq, for each r∈Ar \in A, pq+r≠1p q + r \ne 1 (which is constructively stronger than being prime and minimal among proper ideals). The trivial? antiideal is the ≠\ne-complement of {0}\{0\}.

Given any subset BB of XX, the antisubalgebra generated by BB is the union of all antisubalgebras contained in BB. (This construction, unlike those above, is not predicative.)